Detecting Partial Phase Flip Errors With Ancilla Qubits

• Logical qubits: $\ket{0_{L}} = \ket{+ \ + \ +}$ and $\ket{1_L} = \ket{- \ - \ -}$
• Encode: $\ket{\psi} = \alpha \ket{+ \ + \ +} + \beta{- \ - \ -}$
  • We use this encoding because a complete phase flip would switch $\ket{+}$ to $\ket{-}$

Detection of Complete Phase Flip

• Measure parity of consecutive qubits in the $X$-basis

Example 09.1 (Left-qubit flipped).

Consider the following phase flip:

$$\ket{\psi} = \alpha \ket{- \ + \ +} + \beta \ket{+ \ - \ -} $$

How can we encode the given state?

Partial Phase Flip

• We want to obtain an exact mapping on how partial phase flips change $\ket{\psi}$
• Universal gate operator about $z$-axis with $\vec{n} = \left< 0, 0, 1 \right>$ and $\alpha = \frac{\pi}{2}$